In this paper, the one-dimensional nonlinear Sine–Gordon equation is solved using the “Exponential modified cubic B-spline differential quadrature method with the leave-one-out cross-validation (LOOCV) approach”. By employing the LOOCV approach to determine the optimal value of the parameter \(\epsilon \) involved in the basis function, the accuracy and effectiveness of the results are improved. The combination of this approach with the exponential modified cubic B-spline differential quadrature method, which is novel in the literature, is likely to attract researchers' interest. Additionally, the procedure is implemented on six examples of the Sine–Gordon equation. The results are presented in the form of tables and figures. It is demonstrated that this approach is straightforward and yields superior outcomes compared to the existing literature. This paper also presents an insightful discussion on the significant application of the Sine–Gordon equation in Josephson junctions and its crucial role in new technologies.

本文使用“带有留一交叉验证（LOOCV）方法的指数修正三次B样条微分求积法”来求解一维非线性Sine-Gordon方程。通过采用LOOCV方法确定基函数涉及的参数\(\epsilon \)的最优值，提高了结果的准确性和有效性。这种方法与文献中新颖的指数修正三次B样条微分求积法的结合可能会引起研究人员的兴趣。此外，该过程还在 Sine-Gordon 方程的六个示例上实施。结果以表格和图形的形式呈现。事实证明，与现有文献相比，这种方法简单明了，并且产生了更好的结果。本文还对正弦-戈登方程在约瑟夫森结中的重要应用及其在新技术中的关键作用进行了深入的讨论。